Final answer:
To restore the fundamental mode after doubling the tension of a string, the frequency must be increased by a factor of √(2), because the wave speed is proportional to the square root of the tension.
Step-by-step explanation:
When the tension on a string is doubled, the wave speed in the string increases, because the wave speed (v) is proportional to the square root of the tension over the linear mass density (v ∝ √(μ). Since the fundamental frequency (f1) is equal to the wave speed (v) divided by twice the length of the string (f1 = v/2L), to get the fundamental mode back, the frequency needs to be adjusted to compensate for the change in wave speed.
If the tension is doubled, the new wave speed is √(2) times the original wave speed. Therefore, to achieve the original fundamental frequency, the new frequency required would be f1' = (v √(2) 2L, which is √(2) times the original fundamental frequency. Simply put, the frequency must be multiplied by √(2) to restore the fundamental mode when the tension is doubled.